# Krishna’s Fully Solved B.Sc. Mathematics

SYLLABUS- Solutions B.Sc. MATHEMATICS-IV, Vector Spaces & Matrices

B.A./B.Sc. IV Semester–Paper-I

Vector spaces: Vector space, sub spaces, Linear combinations, linear spans, Sums

and direct sums.

Bases and Dimensions: Linear dependence and independence, Bases and

dimensions, Dimensions and subspaces, Coordinates and change of bases.

Matrices: Idempotent, nilpotent, involutary, orthogonal and unitary matrices, singular

and nonsingular matrices, negative integral powers of a nonsingular matrix Trace of a

matrix.

Rank of a matrix: Rank of a matrix, linear dependence of rows and columns of a

matrix, row rank, column rank, equivalence of row rank and column rank, elementary

transformations of a matrix and invariance of rank through elementary transformations,

normal form of a matrix, elementary matrices, rank of the sum and product of two

matrices, inverse of a non-singular matrix through elementary row transformations

equivalence of matrices.

Applications of Matrices: Solutions of a system of linear homogeneous equations,

condition of consistency and nature of the general solution of a system of linear non-homogeneous

equations, matrices of rotation and reflection.

Real Analysis

B.A./B.Sc. IV Semester–Paper-II

Continuity and Differentiability of functions: Continuity of functions, Uniform

continuity, Differentiability, Taylor's theorem with various forms of remainders.

Integration: Riemann integral-definition and properties, integrability of continuous

and monotonic functions, Fundamental theorem of integral calculus, Mean value

theorems of integral calculus.

Improper Integrals: Improper integrals and their convergence, Comparison test,

Dritchlet’s test, Absolute and uniform convergence, Weierstrass M-Test, Infinite integral

depending on a parameter.

Sequence and Series: Sequences, theorems on limit of sequences, Cauchy’s

convergence criterion, infinite series, series of non-negative terms, Absolute

convergence, tests for convergence, comparison test, Cauchy’s root Test, ratio Test,

Rabbe’s, Logarithmic test, De Morgan’s Test, Alternating series, Leibnitz’s theorem.

Uniform Convergence: Point wise convergence, Uniform convergence, Test of

uniform convergence, Weierstrass M-Test, Abel’s and Dritchlet’s test, Convergence and

uniform convergence of sequences and series of functions.

Mathematical Methods

B.A./B.Sc. IV Semester–Paper-III

Integral Transforms: Definition, Kernel.

Laplace Transforms: Definition, Existence theorem, Linearity property, Laplace

transforms of elementary functions, Heaviside Step and Dirac Delta Functions, First

Shifting Theorem, Second Shifting Theorem, Initial-Value Theorem, Final-Value

Theorem, The Laplace Transform of derivatives, integrals and Periodic functions.

Inverse Laplace Transforms: Inverse Laplace transforms of simple functions,

Inverse Laplace transforms using partial fractions, Convolution, Solutions of differential

and integro-differential equations using Laplace transforms. Dirichlet’s condition.

Fourier Transforms: Fourier Complex Transforms, Fourier sine and cosine

transforms, Properties of FourierTransforms, Inverse Fourier transforms.

**Authors:** A.R Vasishtha

**Date:** 2021

**Upload Date:** 9/29/2021 9:55:13 AM

**Format:** pdf

**Pages:** 397

**OCR:**

**Quality:**

**Language:** English

**ISBN / ASIN:** B0932L1XXV

**ISBN13:**

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